How to determine which model fits my non-linear data the best?

#1
I've got SigmaPlot and JMP. What's the best way to go about determining which curve fits my non-linear data the best? What parameters should I be looking at besides R^2?
 
#2
The best way to determine if the curve fits non-linear data, is to look if it fits. Do the model fit to the data?

What is the model? Do a scatter plot diagram! Does it fit ?
(And don't look at R^2.)
 
#3
SigmaPlot provides a lot of equations that I can use to fit the data. To me, it seems like a lot of them could potentially fit. I was thinking about using AICc but am not sure how to use it in SigmaPlot.